[b]p1.[/b] Alan leaves home when the clock in his cardboard box says $7:35$ AM and his watch says $7:41$ AM. When he arrives at school, his watch says $7:47$ AM and the $7:45$ AM bell rings. Assuming the school clock, the watch, and the home clock all go at the same rate, how many minutes behind the school clock is the home clock? [b]p2.[/b] Compute $$\left( \frac{2012^{2012-2013} + 2013}{2013} \right) \times 2012.$$ Express your answer as a mixed number. [b]p3.[/b] What is the last digit of $$2^{3^{4^{5^{6^{7^{8^{9^{...^{2013}}}}}}}}} ?$$ [b]p4.[/b] Let $f(x)$ be a function such that $f(ab) = f(a)f(b)$ for all positive integers $a$ and $b$. If $f(2) = 3$ and $f(3) = 4$, find $f(12)$. [b]p5.[/b] Circle $X$ with radius $3$ is internally tangent to circle $O$ with radius $9$. Two distinct points $P_1$ and $P_2$ are chosen on $O$ such that rays $\overrightarrow{OP_1}$ and $\overrightarrow{OP_2}$ are tangent to circle $X$. What is the length of line segment $P_1P_2$? [b]p6.[/b] Zerglings were recently discovered to use the same $24$-hour cycle that we use. However, instead of making $12$-hour analog clocks like humans, Zerglings make $24$-hour analog clocks. On these special analog clocks, how many times during $ 1$ Zergling day will the hour and minute hands be exactly opposite each other? [b]p7.[/b] Three Small Children would like to split up $9$ different flavored Sweet Candies evenly, so that each one of the Small Children gets $3$ Sweet Candies. However, three blind mice steal one of the Sweet Candies, so one of the Small Children can only get two pieces. How many fewer ways are there to split up the candies now than there were before, assuming every Sweet Candy is different? [b]p8.[/b] Ronny has a piece of paper in the shape of a right triangle $ABC$, where $\angle ABC = 90^o$, $\angle BAC = 30^o$, and $AC = 3$. Holding the paper fixed at $A$, Ronny folds the paper twice such that after the first fold, $\overline{BC}$ coincides with $\overline{AC}$, and after the second fold, $C$ coincides with $A$. If Ronny initially marked $P$ at the midpoint of $\overline{BC}$, and then marked $P'$ as the end location of $P$ after the two folds, find the length of $\overline{PP'}$ once Ronny unfolds the paper. [b]p9.[/b] How many positive integers have the same number of digits when expressed in base $3$ as when expressed in base $4$? [b]p10.[/b] On a $2 \times 4$ grid, a bug starts at the top left square and arbitrarily moves north, south, east, or west to an adjacent square that it has not already visited, with an equal probability of moving in any permitted direction. It continues to move in this way until there are no more places for it to go. Find the expected number of squares that it will travel on. Express your answer as a mixed number. PS. You had better use hide for answers.

Let $ABCD$ be a convex quadrilateral with $AB = AD$ and $CB = CD$. The bisector of $\angle BDC$ intersects $BC$ at $L$, and $AL$ intersects $BD$ at $M$, and it is known that $BL = BM$. Determine the value of $2\angle BAD + 3\angle BCD$.

Andrew the ant starts at vertex $A$ of square $ABCD$. Each time he moves, he chooses the clockwise vertex with probability $\frac{2}{3}$ and the counter-clockwise vertex with probability $\frac{1}{3}$. What is the probability that he ends up on vertex $A$ after $6$ moves? [i]2015 CCA Math Bonanza Lightning Round #4.3[/i]

Let $n\ge 3$ be a positive integer. In the plane $n$ points which are not all collinear are marked. Find the least possible number of triangles whose vertices are all marked. (Recall that the vertices of a triangle are not collinear.)

Given an equilateral triangle $ABC$, let $M$ be an arbitrary point in space. $(\text{a})$ Prove that one can construct a triangle from the segments $MA, MB, MC$. $(\text{b})$ Suppose that $P$ and $Q$ are two points symmetric with respect to the center $O$ of $ABC$. Prove that the two triangles constructed from the segments $PA,PB,PC$ and $QA,QB,QC$ are of equal area.

Using $ AB$ and $ AC$ as diameters, two semi-circles are constructed respectively outside the acute triangle $ ABC$. $ AH \perp BC$ at $ H$, $ D$ is any point on side $ BC$ ($ D$ is not coinside with $ B$ or $ C$ ), through $ D$, construct $ DE \parallel AC$ and $ DF \parallel AB$ with $ E$ and $ F$ on the two semi-circles respectively. Show that $ D$, $ E$, $ F$ and $ H$ are concyclic.

Convert the following decimal to a common fraction in lowest terms: $ 0.92007200720072007...$ (or $ 0.9\overline{2007}$).

One player conceals a $10$ or $20$ copeck coin, and the other guesses its value. If he is right he gets the coin, if wrong he pays $15$ copecks. Is this a fair game? What are the optimal mixed strategies for both players?

In an acute scalene triangle $ABC$ with incenter $I$, the line $AI$ intersects the circumcircle again at $D$, and let $J$ be a point such that $D$ is the midpoint of $IJ$. Consider points $E$ and $F$ on line $BC$ such that $IE$ and $JF$ are perpendicular to $AI$. Consider points $G$ on $AE$ and $H$ on $AF$ such that $IG$ and $JH$ are perpendicular to $AE$ and $AF$, respectively. Prove that $BG=CH$.

The set of positive nonzero real numbers are partitioned into three mutually disjoint non-empty subsets $(A\cup B\cup C)$. a) show that there exists a triangle of side-lengths $a,b,c$, such that $a\in A, b\in B, c\in C$. b) does it always happen that there exists a right triangle with the above property ?

We consider regular $ n$-gons with a fixed circumference $ 4$. Let $ r_n$ and $ a_n$ respectively be the distances from the center of such an $ n$-gon to a vertex and to an edge. $ (a)$ Determine $ a_4,r_4,a_8,r_8$. $ (b)$ Give an appropriate interpretation for $ a_2$ and $ r_2$ $ (c)$ Prove that $ a_{2n}\equal{}\frac{1}{2} (a_n\plus{}r_n)$ and $ r_{2n}\equal{}\sqrt{a_2n r_n}.$ $ (d)$ Define $ u_0\equal{}0, u_1\equal{}1$ and $ u_n\equal{}\frac{1}{2}(u_{n\minus{}2}\plus{}u_{n\minus{}1})$ for $ n$ even or $ u_n\equal{}\sqrt{u_{n\minus{}2} u_{n\minus{}1}}$ for $ n$ odd. Determine $ \displaystyle\lim_{n\to\infty}u_n$.

Let $n\geq 2$ be an integer. Let us call [i]interval[/i] a subset $A \subseteq \{1,2,\ldots,n\}$ for which integers $1\leq a < b\leq n$ do exist, such that $A = \{a,a+1,\ldots,b-1,b\}$. Let a family $\mathcal{A}$ of subsets $A_i \subseteq \{1,2,\ldots,n\}$, with $1\leq i \leq N$, be such that for any $1\leq i < j \leq N$ we have $A_i \cap A_j$ being an interval. Prove that $\displaystyle N \leq \left \lfloor n^2/4 \right \rfloor$, and that this bound is sharp. (Dan Schwarz - after an idea by Ron Graham)

$x,y\in\left[-\frac{\pi}{4},\frac{\pi}{4}\right],a\in\mathbb{R}$. If $x^3+\sin x-2a=0,4y^3+\sin y \cos y+a=0$, then $\cos (x+2y)=$________.

Let be some rational numbers with the property that their sum, as well as the product of any two of them is integer. Prove that all these are integers.

How many permutations of $123456$ have exactly one number in the correct place?

For fixed coprime positive integers $a,b$, define $n$ to be [i]bad[/i] if it is not of the form $$ax+by,\enspace x,y\in\mathbb N^*$$ Prove that there are finitely many bad positive integers. Also, find the sum of squares of them.

Let $\mathbb{N}^2$ denote the set of ordered pairs of positive integers. A finite subset $S$ of $\mathbb{N}^2$ is [i]stable[/i] if whenever $(x,y)$ is in $S$, then so are all points $(x',y')$ of $\mathbb{N}^2$ with both $x'\leq x$ and $y'\leq y$. Prove that if $S$ is a stable set, then among all stable subsets of $S$ (including the empty set and $S$ itself), at least half of them have an even number of elements. [i]Ashwin Sah and Mehtaab Sawhney[/i]

Find all functions $ f: \mathbb{R}\to\mathbb{R}$ that satisfy the equation: $ f\left(\left(f\left(x\right)\right)^2 \plus{} f\left(y\right)\right) \equal{} xf\left(x\right) \plus{} y$ for any two real numbers $ x$ and $ y$.

We want to find functions $p(t)$, $q(t)$, $f(t)$ such that (a) $p$ and $q$ are continuous functions on the open interval $(0,\pi)$. (b) $f$ is an infinitely differentiable nonzero function on the whole real line $(-\infty,\infty)$ such that $f(0)=f'(0)=f''(0)$. (c) $y=\sin t$ and $y=f(t)$ are solutions of the differential equation $y''+p(t)y'+q(t)y=0$ on $(0,\pi)$. Is this possible? Either prove this is not possible, or show this is possible by providing an explicit example of such $f,p,q$.

Suppose that the 32 computers in a certain network are numbered with the 5-bit integers $00000, 00001, 00010, ..., 11111$ (bit is short for binary digit). Suppose that there is a one-way connection from computer $A$ to computer $B$ if and only if $A$ and $B$ share four of their bits with the remaining bit being $0$ at $A$ and $1$ at $B$. (For example, $10101$ can send messages to $11101$ and to $10111$.) We say that a computer is at level $k$ in the network if it has exactly $k$ 1’s in its label $(k = 0, 1, 2, ..., 5)$. Suppose further that we know that $12$ computers, three at each of the levels $1$, $2$, $3$, and $4$, are malfunctioning, but we do not know which ones. Can we still be sure that we can send a message from $00000$ to $11111$?

A non-right triangle $A_1A_2A_3$ is given. Circles $C_1$ and $C_2$ are tangent at $A_3, C_2$ and $C_3$ are tangent at $A_1$, and $C_3$ and $C_1$ are tangent at $A_2$. Points $O_1,O_2,O_3$ are the centers of $C_1, C_2, C_3$, respectively. Supposing that the triangles $A_1A_2A_3$ and $O_1O_2O_3$ are similar, determine their angles.

It is known that for some $a$ and $b$ the equation $$\frac{x-3}{(x-6)^2} -\frac{x-6}{(x-3)^2} =a(b-9x+x^2)$$ has as its largest root the number $1995$. Find the smallest root of this equation for the same $a$ and $b$.

In a cyclic quadrilateral $ABCD$, the extensions of sides $AB$ and $CD$ meet at point $P$, and the extensions of sides $AD$ and $BC$ meet at point $Q$. Prove that the distance between the orthocenters of triangles $APD$ and $AQB$ is equal to the distance between the orthocenters of triangles $CQD$ and $BPC$.

In the following base-$10$ equation, each of the letter represents a unique digit: $AM\cdot PM=ZZZ$. Find the sum of $A+M+P+Z$. $\text{(A) }15\qquad\text{(B) }17\qquad\text{(C) }19\qquad\text{(D) }20\qquad\text{(E) }21$

Let $a,b,c$ be positive real numbers such that $a^2+b^2+c^2\geqslant 3$. Prove that \[\frac{a^2}{a+b^2}+\frac{b^2}{b+c^2}+\frac{c^2}{c+a^2}\geqslant\frac{3}{2}.\]